Patterns of quantitative genetic variation in multiple dimensions

Abstract

A fundamental question for both evolutionary biologists and breeders is the extent to which genetic correlations limit the ability of populations to respond to selection. Here I view this topic from three perspectives. First, I propose several nondimensional statistics to quantify the genetic variation present in a suite of traits and to describe the extent to which correlations limit their selection response. A review of five data sets suggests that the total variation differs substantially between populations. In all cases analyzed, however, the “effective number of dimensions” is less than two: more than half of the total genetic variation is explained by a single combination of traits. Second, I consider how patterns of variation affect the average evolutionary response to selection in a random direction. When genetic variation lies in a small number of dimensions but there are a large number of traits under selection, then the average selection response will be reduced substantially from its potential maximum. Third, I discuss how a low genetic correlation between male fitness and female fitness limits the ability of populations to adapt. Data from two recent studies of natural populations suggest this correlation can diminish or even erase any genetic benefit to mate choice. Together these results suggest that adaptation (in natural populations) and genetic improvement (in domesticated populations) may often be as much constrained by patterns of genetic correlation as by the overall amount of genetic variation.

Appendix

Our goal is to calculate $\overline{R}$, the average relative selection response to a selection gradient pointing in a random direction and with a fixed magnitude. Selection response is defined here as the length (norm) of $\Updelta \widetilde{\mathbf{z}}$, the vector of proportional changes in the trait means. We will calculate that response relative to a hypothetical unconstrained population whose eigenvalues are all equal to the maximum eigenvalue of the focal population.

The average selection response is

$$ \bar{R} = \frac{{\int_{0}^{\pi } {\int_{0}^{\pi } \ldots {\int_{0}^{2\pi } {R(\Theta )} } } (\frac{dS}{d\Theta })d\theta _{1} d\theta _{2} \ldots d\theta _{n - 1} }}{{\lambda _{1} \int_{0}^{\pi } {\int_{0}^{\pi } \ldots {\int_{0}^{2\pi } {(\frac{dS}{d\Theta })d\theta _{1} d\theta _{2}\ldots d\theta _{n - 1} } } } }} $$

(A1)

The numerator is the expected selection response in the focal population, and the denominator is the response in the hypothetical unconstrained population. R(Θ) is the selection response to a selection gradient oriented in the direction given by the angles in the vector Θ whose elements are θ₁,θ₂,…,θ_n–1. The ratio ($dS/d\Theta$) is the change in surface area of a unit sphere per change in the angles Θ, and is given by

$$ \frac{dS}{d\Theta } = 2\prod\limits_{i = 1}^{n - 2} {\sin ^{i} \theta _{i} } $$

(A2)

(see http://en.wikipedia.org/wiki/Hypersphere).

Without loss of generality, we can choose coordinates that diagonalize the genetic covariance matrix. Then the eigenvalues are equal to the genetic variances (ordered from largest to smallest). The magnitude of the selection response to a given selection gradient β is

$$ R = \left[\sum\limits_{i = 1}^{n} {(\lambda _{i} \beta _{i})^{2} } \right]^{1/2} $$

(A3)

where $\lambda _{i}$ is again the ith eigenvalue and $\beta _{i}$ is the element of the selection gradient corresponding to that trait. For a selection gradient of unit length, R can be converted to the polar coordinates of Eq. A1 using

$$ \beta _{i} = \cos _{i,n}^{*} (\theta _{i} )\prod\limits_{j = 1}^{i - 1} {\sin (\theta _{j} )} $$

(A4)

where

$$ \cos _{i,n}^{*} (\theta _{i} ) = \left\{ {\begin{array}{*{20}c} {\cos (\theta _{i} )\quad {\text{if}}\,i \ne n} \\ {1\quad\quad {\text{if}}\,i = n} \\ \end{array} } \right. $$

(A5)

Substituting Eqs. A2–A5 into (A1) gives the average selection response $\overline{R}$ in terms of integrals that can be evaluated numerically once the eigenvalues are specified.

I calculated $\overline{R}$ this way using Mathematica (Wolfram 2003) for the three scalar-valued data sets. The results are shown in Table 2. For the function-valued traits, the average selection response is 0 because only a finite number of eigenvalues are positive, but there are an infinite number of trait combinations on which selection could theoretically act.

To better understand how the distribution of eigenvalues affects $\overline{R}$, I then considered hypothetical populations in which the eigenvalues decline geometrically (exponentially). The ratio of successive eigenvalues of the standardized genetic covariance matrix, which is constant, is denoted k. Thus k = 1 is the case where all traits have equal genetic variance and no correlation, while if k = 0 all genetic variation lies along a single dimension. It is convenient to set the leading eigenvector to λ₁ = 1. The value of the ith eigenvalue is then

$$ \lambda _{i} = k_{{}}^{i - 1} \lambda _{1}^{{}} . $$

(A6)

Substituting that expression into (A3) and then numerically integrating (A1) using Mathematica (Wolfram 2003) gives the results shown in Fig. 3.

It is possible to get simple analytic expressions for two special cases. With k = 1, the population has no constraints, and one can show Eq. A1 is equal to unity (as it must). At the other extreme, consider the case of k = 0, so that all genetic variation lies in a single dimension. Then we get

$$ \begin{aligned} \bar{R} &= \frac{{\int_{0}^{\pi } {\int_{0}^{\pi } \ldots {\int_{0}^{2\pi } {|\cos \theta _{1} |(\prod\limits_{k = 1}^{n - 2} {\sin ^{k} \theta _{k} } )} } } d\theta _{1} d\theta _{2} \ldots d\theta _{n - 1} }}{{\int_{0}^{\pi } {\int_{0}^{\pi } \ldots{\int_{0}^{2\pi } {(\prod\limits_{k = 1}^{n - 2} {\sin ^{k} \theta _{k} } )} } } d\theta _{1} d\theta _{2} \ldots d\theta _{n - 1} }}\\ &= \frac{2\Upgamma (n/2)}{(n - 1)\sqrt \pi \Upgamma ((n - 1)/2)} \end{aligned} $$

(A7)

where Γ() is the gamma function. This appears in the text as Eq. 5.